Line up everyone with £1,000-worth of Premium Bonds in order of their year’s winnings, and the person half way along would have won … not a penny! In fact, you’d need to walk past two-thirds of the line until you hit the first £25 winner.

NS&I’s Premium Bonds are the UK’s single most popular savings product, with £47 billion invested in them. Yet the rates are low and the safety advantage negligible – so are they worth it?

**Doing the numbers **

The nearest thing Premium Bonds have to an interest rate is its annual prize rate, currently 1.3pc. This describes the ‘‘average’’ payout, but it is just a vague watermark. It describes the mean average, indicating that for every £100 paid into bonds, on average £1.30 a year is paid out – yet in practice this is impossible, as the smallest prize is £25. In fact if 20 people each had £100 invested, for one to win £25-plus, the remaining 19 would have to win nothing.

A far better indication of what someone with typical luck would win is the ‘‘the person half-way along’’ measure. Those who can dredge up their school maths will remember this is called the median average.

To best demonstrate this, here’s an extreme example.

Imagine I sold a million people a £1 lottery ticket, and then paid just one winner a million pounds.

I could argue, mathematically, that the (mean) average payout was £1, so on average everyone got their money back. This, of course is bonkers. Almost everyone wins nothing – which is the result you’d discover if you lined them all up and asked the mid-way person.

The Premium Bond prize rate is a lesser version of this: to pay a million-pound winner, many thousands need to earn nothing.

**The Premium Bond Probability Calculator **

To accurately calculate the odds, you need to use something called “multinomial probability”. After all, to work out the chance of someone winning £200 a year, they could win 2 x £100, 8 x £25, 4 x £50, or a host of other variants. This multitude of probabilities means accurate calculation is hellish.

A few years ago I set myself a challenge to do it. I failed. I got one of my team with a top maths degree to try. He failed. We contacted a London School of Economics (LSE) Professor of Financial Mathematics – she knew how to work it out, but she needed a specialist to do it for her.

Eventually we tracked down a post-doctoral cosmology statistician (someone who calculates star movements) who had the requisite skills, and he wrote us an algorithm to build PremiumBondCalculator.com. This allows you to plug in how many bonds you have, and it will predict your likely winnings. It proves that at every value someone with typical luck will earn less than the quoted prize rate.

**Don’t think of it as winning **

The most powerful psychological sell of Premium Bonds is that interest is called “winnings”. This lottery-effect hooks you into the unlikely dream of bagging a million-pound prize.

I often hear excited comments such as: “My friend wins £25 every few months!” Yet someone with £10,000-worth of bonds should win £100 a year – that’s £25 every few months. The same money in a top cash Isa would ‘‘win’’ £175 a year, guaranteed.

Normally in finance there’s a risk premium: you should expect higher returns when there’s a higher risk. For some perverse reason, with Premium Bonds people tend to be happy to earn less when there’s a risk.

**Premium Bonds versus savings **

In the battle of tax-free savings, even the bloated 1.3pc tax-free prize rate compares poorly with the top 1.75pc easy-access cash Isa.

Nor does it do well against Santander’s 123 bank account, which can pay 3pc interest, or 2.4pc after tax at the basic rate, 1.8pc at the higher rate (it does have a £2 per month fee, but the cashback paid on bills usually more than covers it).

Only against the top normal savings, which have just dropped to a pitiful 1.5pc, does it look a winner, as basic-rate taxpayers will only earn 1.2pc, and higher rate 0.9pc. Yet the Probability Calculator shows a truer picture.

**Premium Bonds v Savings**

**Are Premium Bonds worth it? **

As chance is involved, there’s no definitive answer. My talk of typical luck ignores the fact that some will always beat the odds. How you value ‘‘the dream’’ is up to you.

Based on the odds, Premium Bonds are certainly a good bet for higher and top-rate taxpayers who’ve filled their cash Isa and have exhausted quirky savings like Santander’s 123 account, or First Direct’s 6pc regular saver (see www.mse.me/savings).

Even for basic rate taxpayers, with the top deals now so weak, Premium Bonds look an OK bet. But this is unusual. That’s likely to reverse, either as savings rates rise, or Premium Bond rates drop – as NS&I isn’t allowed to dominate the market. Normally, Premium Bond odds of beating savings are minuscule and they should be avoided with all but a non-significant portion of your assets.

So while Premium Bonds aren’t all-star winners, they’re looking as good as they ever have. Though if it’s the million you really want – for better odds, put your money in a top savings account, then use the interest to buy lottery tickets.

**Quick facts on Premium Bonds **

**• How do they work? **They’re a savings account where the interest earned is decided by a monthly prize draw. Each £1 buys a bond entered into the draw. Winnings are tax-free.

**• How much can I put in them? **The minimum is £100 (or £50 for monthly standing orders), the maximum £30,000. Though some with older holdings have as little as £1.

**• How do I buy them? **At nsandi.com, in Post Offices, by phone – on 0500 500 000 – or by post.

**• Does each bond have an equal chance? **Yes. NS&I sexes things up by the personification of the IT equipment, Ernie (Electronic Random Number Indicator Equipment). Yet it’s an audited, random-number process. Urban myths suggest older bonds have less chance. Not true – though some people remain unconvinced, I suspect because inflation eats away at older holdings, so they’re smaller and therefore win less.

**• Can winnings be reinvested?** Yes. You can opt to have winnings automatically reinvested into more bonds up to the maximum of £30,000.

**• What are the odds of winning? **The huge majority of prizes, well over 1.7 million, are of £25; 12,000 are of £50 and £100 and the remaining 3,000-odd are for £500 or more. Only one is for £1 million. The odds of winning the top prize for each bond are 1 in 47 billion (a lottery ticket has odds of 1 in 14 million for the jackpot, though you don’t retain the capital). Slightly worse odds than a bookie would give you for an alien landing in Trafalgar Square then tossing a coin that lands on its edge.

**• I’m sure I had old bonds, how do I trace them?** Mylostaccount.org.uk enables you to do this free. Go to nsandi.com to check if you’ve an unclaimed prize (there are £46million-worth).

**• When can I take my money out? **At any time, though it can take up to three days for phone and internet applications, and eight days for postal.

**• Are Premium Bonds safer than savings? **They are operated by the state-backed NS&I, so as safe as it gets. That’s far less of an advantage than it was: the maximum you can put in is £30,000, yet since 2011 every UK-regulated savings account (see mse.me/safesavings) is guaranteed up to £85,000 per person via the Financial Services Compensation Scheme. You could argue, NS&I should never go bust (unless the UK does), while a bank may do, and you’d wait for compensation. Yet for practical purposes this is likely to make only a negligible difference.

*Martin Lewis is the creator of moneysavingexpert.com. You can join the 8m people in the UK who get his weekly tips emailed at www.moneysavingexpert.com/latesttips *